Importance Sampling Spherical Harmonics

In Computer Graphics Forum (Proceedings of Eurographics), 2009

In this paper we present a method for importance sampling spherical harmonic functions such as the BRDF on the left. Our technique can also be combined with wavelets to importance sample the product of a spherical harmonic function and a Haar wavelet function (right).

Abstract

In this paper we present the first practical method for importance sampling functions represented as spherical harmonics (SH). Given a spherical probability density function (PDF) represented as a vector of SH coefficients, our method warps an input point set to match the target PDF using hierarchical sample warping. Our approach is efficient and produces high quality sample distributions. As a by-product of the sampling procedure we produce a multi-resolution representation of the density function as either a spherical mip-map or Haar wavelet. By exploiting this implicit conversion we can extend the method to distribute samples according to the product of an SH function with a spherical mip-map or Haar wavelet. This generalization has immediate applicability in rendering, e.g., importance sampling the product of a BRDF and an environment map where the lighting is stored as a single high-resolution wavelet and the BRDF is represented in spherical harmonics. Since spherical harmonics can be efficiently rotated, this product can be computed on-the-fly even if the BRDF is stored in local-space. Our sampling approach generates over 6 million samples per second while significantly reducing precomputation time and storage requirements compared to previous techniques.

Cite

@article{jarosz09importance,
author = "Jarosz, Wojciech and Carr, Nathan A. and Jensen, Henrik Wann",
title = "Importance Sampling Spherical Harmonics",
journal = "Computer Graphics Forum (Proceedings of Eurographics)",
volume = "28",
number = "2",
year = "2009",
month = "apr",
pages = "577–586",
address = "Munich, Germany",
doi = "10.1111/j.1467-8659.2009.01398.x",
abstract = "In this paper we present the first practical method for importance sampling functions represented as spherical harmonics (SH). Given a spherical probability density function (PDF) represented as a vector of SH coefficients, our method warps an input point set to match the target PDF using hierarchical sample warping. Our approach is efficient and produces high quality sample distributions. As a by-product of the sampling procedure we produce a multi-resolution representation of the density function as either a spherical mip-map or Haar wavelet. By exploiting this implicit conversion we can extend the method to distribute samples according to the product of an SH function with a spherical mip-map or Haar wavelet. This generalization has immediate applicability in rendering, e.g., importance sampling the product of a BRDF and an environment map where the lighting is stored as a single high-resolution wavelet and the BRDF is represented in spherical harmonics. Since spherical harmonics can be efficiently rotated, this product can be computed on-the-fly even if the BRDF is stored in local-space. Our sampling approach generates over 6 million samples per second while significantly reducing precomputation time and storage requirements compared to previous techniques."
}